CS 3100, 11 / 23/ 10 Ganesh Gopalakrishnan

1. http://www.cs.utah.edu/fv CS 3100, 11/23/10 Ganesh Gopalakrishnan

2. Asg8 • 1(f) – convert <==> to => • All problems: assume x,y,k are in Nat • a,b,c are of course Boolean • I misspoke about mapping reductions • They need not be 1-1 • GCD questions: follow defn of GCD • Is a divisor • Is the largest • X and Y divisible by Z means (X+/-Y) div by Z • Clique questions : Think of how cliques are built • What is a 1-clique? 2-clique? 3-clique? 4-clique? … • Do Qn4 without using Rice’s Theorem • Similar to Reg_TM problem • “Floor trap-door is opened” based on whether M accepts w

3. Asg8 • Counting Boolean functions over N inputs • Of course, only finitely many • But grows quite fast! • Contrast with counting Nat -> Nat functions • Try to enumerate functions • We can find a function not in the enumeration • Is of higher cardinality

4. Mapping reductions • Basic idea: • Given a set A and a Set B, we are seeking an “embedding of A in B” that • Preserves membership • A <=m B is the notation • You can read it also as “A is less hard or the same hardness as B” • We are going to practice it on 2(a) and 2(b) – no computability connotation • Simply try to read “IFF” • Then do 2(c) which tries to force you to think of language -> language mapping redns • <M,w> pairs in A_TM are mapped to <M> singletons in the language A_bt • See if all conditions for an MR are satisfied by the constructed mapping reduction

5. Mapping reductions • Given an M and w • Build a new TM M_w that has “w” embedded in it • Say in a “data array” • Then give M_w to the claimed decider for A_bt • What will M_w do when run? • Erases input • Writes w from data array onto tape • Runs M’s code on input • If D_A_bt can take machines in an “unsuspecting” manner and claim to answer the acceptance of “e” of those machines • Then it may be fed a “loaded” machine such as M_w

6. Mapping reductions • Study mapping reduction in the case of NPC (3CNF formula to Graphs) also • Preserves hardness in both cases • If we can solve A_bt, we can solve A_TM because A_TM is <= in hardness • If we can solve Clique in poly-time, we can solve 3SAT also in poly-time

7. MT2 • Language blending • S -> 0S | 1S | e | T • T -> generates a CFG but its structure is blended away! Try this: S -> T T | U U -> 0 U 0 0 | # T -> 0 T | T 0 | #

8. Complexity theory • Various complexity classes • Reduction principles remain the same • Exp-time complete • P-space complete • Pspace and Npspace are the same • Space can be reused! Time can’t be! • How about energy? • Charles Seitz and Tom McKnight (and others) used to talk about “Hot clocking” and “Adiabatic circuits” • Charge sloshes back and forth (inductor in clock path; circuit is capacitive) • Some energy recovery happens – as opposed to this, in real CMOS ckts, the energy pumped into the capacitors is destroyed and turned into heat • So I don’t know whether the “reuse” of energy happens in the same sense • Google queries : each can heat a cup of water to near boil • But the water in the hydro plant would otherwise have hit the rocks and generated heat that way also • Bottomline: if you harvest energy at every spot, perhaps we are OK burning a whole lot (roads and roofs can produce energy)

9. Complexity theory • NP-complete • Ptime and Nptime are different • NP-hard • P-complete • Relevant for parallelization • BFS can be parallelized more easily • DFS – not so • Is P-complete

10. Complexity theory • Sometimes, complexity classes are not known • E.g. for some problems, the time-complexity characterization is still an open problem • In that case, just do what we can! i.e. get space complexity results • NP-hard : At least as hard as NP • All problems in NP have a <=m to that problem which is NPH • Note that Diophantine is NPH • At least as hard as NP • But really really really hard (undecidable) • So to show NPC , must show that it is in NP also • ND algorithm has a P-time solution

11. Complexity theory • ND algorithm • Guess and check • Guess must result in poly-long “certificate” • Check must be doable in poly-time • Showing that some problems have poly certificates took effort! • Pratt showed that Primality certificates are poly (in 1976) • But then we have a cool result: If NPC and CO-NP then NP = Co-NP • But since the consequent is unlikely, then for problems that are NP and Co-NP, then it may be that they are not NPC • Sure enough, Agrawal, Kayal, and Saxena (the latter two are BS CS students!) showed that primarily has a Det Poly checking algorithm • This is NOT the same as prime factorization : the language changes!

12. Complexity theory • The same happened to lin programming • Kachian came up with Poly algorithm • But it was well known that Lin Prog and its complement are in NP • (there is more to this… ask Prof. Suresh Venkat) • Certificate “blowup” is indicative of hardness • You saw that in PCP and also in Diophantine in a different light (not having succinct certificates is trouble)

13. Complexity theory • Strongly NPC • Problem hardness does not change by encoding method • 3SAT, Tetris, etc are so • 3-partitioning is so • Not strongly NPC (pseudo-polynomial) • Can reduce complexity by bloating input • 2-partitioning is so

14. NPC uses • Don’t run away if NPC • Don’t run away if undecidable • All it means is that the FULL language is hard • Pieces of the language may be easy • That is what BDDs will sort of teach us • Will do this + Bool Sat after Turkey-Day • Gobble Gobble meanwhile!

15. Wish you…